Recognizing the Graph of a Quadratic Equation in Two Variables

We have graphed equations of the form \(Ax+By=C\). We called equations like this linear equations because their graphs are straight lines.

Now, we will graph equations of the form \(y=a{x}^{2}+bx+c\). We call this kind of equation a quadratic equation in two variables.

Quadratic Equation in Two Variables

A quadratic equation in two variables, where \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\) are real numbers and \(a\ne 0\), is an equation of the form

\(y=a{x}^{2}+bx+c\)

Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations.

Let’s look first at graphing the quadratic equation \(y={x}^{2}\). We will choose integer values of \(x\) between \(-2\) and 2 and find their \(y\) values. See the table below.

\(y={x}^{2}\)
\(x\)	\(y\)
0	0
1	1
\(-1\)	1
2	4
\(-2\)	4

Notice when we let \(x=1\) and \(x=-1\), we got the same value for \(y\).

\(\begin{array}{cccc}y={x}^{2}\hfill & & & y={x}^{2}\hfill \\ y={1}^{2}\hfill & & & y={\left(-1\right)}^{2}\hfill \\ y=1\hfill & & & y=1\hfill \end{array}\)

The same thing happened when we let \(x=2\) and \(x=-2\).

Now, we will plot the points to show the graph of \(y={x}^{2}\). See the figure below.

This figure shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (0, 0). Other points on the curve are located at (-2, 4), (-1, 1), (1, 1) and (2, 4).

The graph is not a line. This figure is called a parabola. Every quadratic equation has a graph that looks like this.

In the example above you will practice graphing a parabola by plotting a few points.

Example

Graph \(y={x}^{2}-1\).

Solution

We will graph the equation by plotting points.

Choose integers values for x, substitute them into the equation and solve for y.
Record the values of the ordered pairs in the chart.
Plot the points, and then connect them with a smooth curve. The result will be the graph of the equation\(y={x}^{2}-1\).

How do the equations \(y={x}^{2}\) and \(y={x}^{2}-1\) differ? What is the difference between their graphs? How are their graphs the same?

All parabolas of the form \(y=a{x}^{2}+bx+c\) open upwards or downwards. See the figure below.

Notice that the only difference in the two equations is the negative sign before the \({x}^{2}\) in the equation of the second graph in the figure above. When the \({x}^{2}\) term is positive, the parabola opens upward, and when the \({x}^{2}\) term is negative, the parabola opens downward.

Parabola Orientation

For the quadratic equation \(y=a{x}^{2}+bx+c\), if:

The image shows two statements. The first statement reads “a greater than 0, the parabola opens upwards”. This statement is followed by the image of an upward opening parabola. The second statement reads “a less than 0, the parabola opens downward”. This statement is followed by the image of a downward opening parabola.

Example

Determine whether each parabola opens upward or downward:

(a) \(y=-3{x}^{2}+2x-4\)
(b) \(y=6{x}^{2}+7x-9\)

Solution

(a) Find the value of "a".		Since the “a” is negative, the parabola will open downward.
(b) Find the value of "a".		Since the “a” is positive, the parabola will open upward.

Recognizing the Graph of a Quadratic Equation in Two Variables